Delay effect on phase transitions in traffic dynamics

A dynamical model of traffic is proposed to take into account the effect of acceleration delay. In the limit of no delay, the model reproduces the optimal velocity model of traffic. When the delay is small, it is shown that the phase transition among the freely moving phase, the coexisting phase, and the uniform congested phase occurs below the critical point. Above the critical point, no phase transition occurs. The value ${a}_{c}$ of the critical point increases with increasing delay time $1/b,$ where $a$ is the friction coefficient (or sensitivity parameter). When the delay time is longer than $\frac{1}{2},$ the critical point disappears and the phase transition always occurs. The linear stability theory and nonlinear analysis are applied. The critical point predicted by the linear stability theory agrees with the simulation result. The modified Korteweg--de Vries (KdV) equation is obtained from the nonlinear analysis near the critical point. The phase separation line obtained from the modified KdV equation is consistent with the simulation result.